Estimating frequency-offsets and multi-antenna channels in MIMO OFDM systems

ABSTRACT

Techniques are described for carrier frequency offset (CFO) and channel estimation of orthogonal frequency division multiplexing (OFDM) transmissions over multiple-input multiple-output (MIMO) frequency-selective fading channels. A wireless transmitter forms blocks of symbols by inserting training symbols within two or more blocks of information-bearing symbols. The transmitter applies a hopping code to each of the blocks of symbols to insert a null subcarrier at a different position within each of the blocks of symbols, and a modulator outputs a wireless signal in accordance with the blocks of symbols. A receiver receives the wireless signal and estimates the CFO, and outputs a stream of estimated symbols based on the estimated CFO.

This application is a continuation of U.S. application Ser. No.13/777,993, filed Feb. 26, 2013, which is a continuation of U.S.application Ser. No. 13/301,482, filed Nov. 21, 2011, which is acontinuation of U.S. application Ser. No. 10/850,961, filed May 21, 2004(now U.S. Pat. No. 8,064,528), which claims the benefit of U.S.Provisional Application Ser. No. 60/472,297, filed May 21, 2003, theentire content of each being incorporated herein by reference.

STATEMENT REGARDING FEDERALLY SPONSORED RESEARCH OR DEVELOPMENT

This invention was made with government support under CCR-0105612awarded by the National Science Foundation and DAAD19-01-2-0011 awardedby the Army Research Lab (ARL/CTA). The government has certain rights inthe invention.

TECHNICAL FIELD

The invention relates to communication systems and, more particularly,carrier frequency offset estimation and channel estimation incommunication systems.

BACKGROUND

Providing reliable high data rate services, e.g. real-time multimediaservices, over wireless communication channels is a paramount goal indeveloping coding and modulation schemes. When a data rate for wirelesscommunication systems is high in relation to bandwidth, multipathpropagation may become frequency-selective and cause intersymbolinterference (ISI). Multipath fading in wireless communication channelscauses performance degradation and constitutes the bottleneck forincreasing data rates.

Orthogonal frequency division multiplexing (OFDM) is inherentlyresistant to multipath fading and has been adopted by many standardsbecause it offers high data-rates and low decoding complexity. Forexample, OFDM has been adopted as a standard for digital audiobroadcasting (DAB) and digital video broadcasting (DVB) in Europe andhigh-speed digital subscriber lines (DSL) in the United States. OFDM hasalso been proposed for local area mobile wireless broadband standardsincluding IEEE 802.11a, IEEE 802.11g, MMAC and HIPERLAN/2. Additionally,space-time (ST) multiplexing with multiple antenna arrays at both thetransmitter and receiver are effective in mitigating fading andenhancing data rates. Therefore, multi-input multi-output (MIMO) OFDM isattractive for multi-user wireless communication systems. However, MIMOOFDM systems have increasing channel estimation complexity as the numberof antennas increases due to the increased number of unknowns which mustbe estimated and have great sensitivity to carrier frequency offsets(CFO).

Typical single-input single-output (SISO) OFDM systems rely on blocks oftraining symbols or exploit the presence of null sub-carriers in orderto acquire channel state information (CAI) to mitigate CFO and performchannel estimation. In the IEEE 802.11a, IEEE 802.11g, and HIPERLAN/2standards, sparsely placed pilot symbols are present in every OFDMsymbol and pilot symbols are placed in the same positions from block toblock. Additionally, channel estimation is performed on a per blockbasis.

For channel state information (CSI) acquisition, three classes ofmethods are available: blind methods which estimate CSI solely from thereceived symbols; differential methods that bypass CSI estimation bydifferential encoding; and input-output methods which rely on trainingsymbols that are known a priori to the receiver. Relative to trainingbased schemes, differential approaches incur performance loss by design,while blind methods typically require longer data records and entailhigher complexity. Although training methods can be suboptimal and arebandwidth consuming, training methods remain attractive in practicebecause they decouple symbol detection from channel estimation, therebysimplifying receiver complexity and relaxing the requiredidentifiability conditions.

SUMMARY

In general, the invention is directed to techniques for carrierfrequency offset (CFO) and channel estimation of orthogonal frequencydivision multiplexing (OFDM) transmissions over multiple-inputmultiple-output (MIMO) frequency-selective fading channels. Inparticular, techniques are described that utilize training symbols suchthat CFO and channel estimation are decoupled from symbol detection atthe receiver. Unlike conventional systems in which training symbols areinserted within a block of space-time encoded information-bearingsymbols to form a transmission block, the techniques described hereininsert training symbols over two or more transmission blocks.Furthermore, training symbols may include both non-zero symbols and zerosymbols and are inserted so that channel estimation and CFO estimationare performed separately. Zero symbols, referred to as null subcarriers,are utilized that change position, i.e. “hop”, from block to block. Inthis manner, the information-bearing symbols and training symbols arereceived in a format in which the training symbols are easily separatedfrom the information-bearing symbols, thereby enabling CFO estimation tobe performed prior to channel estimation.

In one embodiment, the invention is directed to a method comprisingforming blocks of symbols by inserting training symbols within two ormore blocks of information-bearing symbols; applying a hopping code toeach of the blocks of symbols to insert a null subcarrier at a differentposition within each of the blocks of symbols; and outputting wirelesstransmission signal in accordance with the blocks of symbols.

In another embodiment, the invention is directed to a method comprisingreceiving a wireless signal transmitted from a stream of blocks ofsymbols, wherein each block of symbols includes one or moreinformation-bearing symbols, one or more training symbols, and at leastone null subcarrier at a different position within each of the blocks ofsymbols. The method further comprises outputting estimated symbols basedon the received wireless signal.

In another embodiment, the invention is directed to a wirelesscommunication device comprising a training symbol insertion module toform blocks of symbols by inserting training symbols within two or moreblocks of information-bearing symbols, wherein the training symbolinsertion module applies a hopping code to each of the blocks of symbolsto insert a null subcarrier at a different position within each of theblocks of symbols; and a modulator to output a wireless transmissionsignal in accordance with the blocks of symbols.

In another embodiment, the invention is directed to a wirelesscommunication device comprising: one or more antennas that receive awireless signal transmitted from a stream of blocks of symbols, whereineach block of symbols includes one or more information-bearing symbols,one or more training symbols, and at least one null subcarrier at adifferent position within each of the blocks of symbols; an carrierfrequency offset estimator to estimate a carrier frequency offset of thereceived signal based on the positions of the null subcarriers; and adecoder to output a stream of estimated symbols based on the receivedwireless signal and the estimated carrier frequency offset.

In another embodiment, the invention is directed to a computer-readablemedium containing instructions. The instructions cause a programmableprocessor to form blocks of symbols by inserting training symbols withintwo or more blocks of information-bearing symbols; apply a hopping codeto each of the blocks of symbols to insert a null subcarrier at adifferent position within each of the blocks of symbols; and outputwireless transmission signal in accordance with the blocks of symbols.

The described techniques may offer one or more advantages. For example,instead of performing CFO and MIMO channel estimation on a per blockbasis, several transmission blocks are collected by a receiver forestimating CFO and the MIMO frequency-selective channels, therebyresulting in an efficient use of bandwidth. Further, because thetraining symbols are inserted in a manner that decouples CFO and channelestimation from symbol detection, low-complexity CFO and channelestimation can be performed. Moreover, the described techniques allowfor full acquisition range of the CFO estimator and identifiability ofthe MIMO channel estimator.

Other advantages of performing block equalization may include improvedbit-error-rate (BER) performance relative to typical techniques andflexibility to adjust the number of blocks collected to perform channelestimation. Because of the improved BER performance, less expensivevoltage controlled oscillators may be used. Additionally, the trainingpatterns of the described techniques can easily be implemented bycurrent OFDM standards, such as IEEE 802.11a and IEEE 802.11g.

The details of one or more embodiments of the invention are set forth inthe accompanying drawings and the description below. Other features,objects, and advantages of the invention will be apparent from thedescription and drawings, and from the claims.

BRIEF DESCRIPTION OF DRAWINGS

FIG. 1 is a block diagram illustrating an exemplary wireless multi-usercommunication system in which multiple transmitters communicate withmultiple receivers through a wireless communication channel.

FIG. 2 is a block diagram illustrating in further detail one embodimentof a transmitter and a receiver within the multi-user communicationsystem of FIG. 1.

FIG. 3 illustrates example transmission blocks generated by thetransmitter of FIG. 2.

FIG. 4 is a flowchart illustrating an example mode of operation of thecommunication system of FIG. 2 in which a receiver performs CFOestimation and channel estimation on an OFDM transmission signal outputby a transmitter.

FIGS. 5-12 are graphs illustrating performance estimates of the CFO andchannel estimation techniques described herein.

DETAILED DESCRIPTION

Throughout the Detailed Description bold upper letters denote matrices,bold lower letters stand for column vectors, (•)^(T) and (•)^(H) denotetranspose and Hermitian transpose, respectively; (•)* denotes conjugateand └•┐ denotes the nearest integer. E[•] stands for expectation anddiag[x] stands for a diagonal matrix with x on its main diagonal; matrixD_(N)(h) with a vector argument denotes an N×N diagonal matrix withD_(N)(h)=diag[h]. For a vector, ∥•∥ denotes the Euclidian norm.[A]_(k,m) denotes the (k, m)th entry of a matrix A, and [x]_(m), denotesthe mth entry of the column vector x; I_(N) denotes the N×N identitymatrix; e_(i) denotes the (i+1)st column of I_(N);[F_(N)]_(m,m)=N^((1/2))exp(−j2Πmn|N) denotes the N×N fast fouriertransform (FFT) matrix; and we define f:=[1, exp(jω), exp(j(N−1)ω)^(T).

FIG. 1 is a block diagram illustrating a multi-user wirelesscommunication system 2 in which multiple transmitters communicate withmultiple receivers 6 through wireless communication channel 8. Ingeneral, the invention describes techniques for performing carrierfrequency offset (CFO) and channel estimation of orthogonal frequencydivision multiplexing (OFDM) transmissions output by transmitters 4 overmultiple-input multiple-output (MIMO) frequency-selective fading channel8. As described herein, the techniques maintain orthogonality amongsubcarriers of OFDM transmissions through channel 8 allowinglow-complexity receivers 6 and full acquisition range of the CFO.

Transmitters 4 output a transmission signal in accordance with a blockof symbols in which two or more training symbols are inserted and inwhich a hopping code is applied. A block of training symbols includingtwo or more training symbols may be inserted within a block ofspace-time encoded information-bearing symbols. A hopping code may thenbe applied to the resulting block of symbols to insert a nullsubcarrier, i.e. zero symbol, within the block symbols such that thenull subcarrier changes position, i.e. “hops”, from block to block.Unlike conventional systems in which training symbols are insertedwithin a single transmission block, the techniques described hereininsert training symbols over two or more transmission blocks.Consequently, transmitters 4 may insert a sequence of training symbolsover two or more transmission blocks, thereby increasing bandwidthefficiency because smaller blocks of training symbols may be used.Receivers 6 may then collect the training symbols inserted within thetwo or more transmission blocks in order to perform channel estimation.Furthermore, the information-bearing symbols and training symbols arereceived through communication channel 8 by receivers 6 in a format inwhich the training symbols are easily separated from theinformation-bearing symbols, thereby enabling CFO estimation to beperformed prior to channel estimation. As a result, the techniquesdescribed herein may have improved bit-error-rate (BER) performance overconventional alternatives.

The described techniques can work with any space-time encodedtransmission and is backwards compatible with OFDM which has beenadopted as a standard for digital audio broadcasting (DAB) and digitalvideo broadcasting (DVB) in Europe and high-speed digital subscriberlines (DSL) in the United States. OFDM has also been proposed for localarea mobile wireless broadband standards including IEEE 802.11a, IEEE802.11g, MMAC and HIPERLAN/2.

The techniques described herein apply to uplink and downlinktransmissions, i.e., transmissions from a base station to a mobiledevice and vice versa. Transmitters 4 and receivers 6 may be any deviceconfigured to communicate using a multi-user wireless transmissionincluding a cellular distribution station, a hub for a wireless localarea network, a cellular phone, a laptop or handheld computing device, apersonal digital assistant (PDA), a Bluetooth™ enabled device, and otherdevices.

FIG. 2 is a block diagram illustrating in further detail the multi-usercommunication system of FIG. 1. In particular, FIG. 2 illustratesexemplary embodiments of multi-antenna transmitter 4 and multi-antennareceiver 6 communicating over MIMO frequency-selective channel 8 in thepresence of a CFO. Multi-antenna transmitter 4 and multi-antennareceiver 6 have N_(t) and N_(r) antennas, respectively. While OFDMtransmissions are inherently resilient to multipath fading, OFDMtransmissions are more sensitive to frequency offsets than singlecarrier systems. Frequency offsets can occur when a voltage controlledoscillator (VCO) of receiver 6 is not oscillating at exactly the samecarrier frequency as a VCO of transmitter 4 and can also occur as aresult of the Doppler effect. When the frequency offset is permanent, itis typically referred to as a carrier frequency offset and when thefrequency offset varies over time, it is typically referred to as phasenoise. Frequency offsets cause a degradation in BER performance becausethe orthogonality among subcarriers is destroyed and the subcarriers caninterfere with each other.

Generally, receiver 6 corresponds to a particular user performing CFOand channel estimation of OFDM transmissions output by transmitter 4through MIMO frequency-selective fading channel 8 in the presence of aCFO. Each information-bearing symbol s(n) 10 is selected from a finitealphabet and input into serial to parallel converter (S/P) 11 whichparses N_(s) information-bearing symbols from a serial stream ofinformation-bearing symbols into blocks of information-bearing symbols.The nth entry of the kth block of the block of information-bearingsymbols is denoted [s(k)]_(n)=s(kN_(s)+n). Space-Time coder 13 encodesand/or multiplexes each block s(k) in space and time to yield blocks{c_(μ)(k)}_(μ=1) ^(N) ^(t) 14 of length N_(c). Space-Time coder 13 mayapply any space-time code to yield blocks {c_(μ)(k)}_(μ=1) ^(N) ^(t) 14for each respective transmit antenna of multi-antenna transmitter 4.

Each of training symbol insertion units 15 inserts two or more trainingsymbols, which may have non-zero or zero values, within space-timeencoded blocks {c_(μ)(k)}_(μ=1) ^(N) ^(t) 14 and applies a hopping codeto blocks {c_(μ)(k)}_(μ=1) ^(N) ^(t) 14 to form a vectors ū_(μ)(k) 16with length N for the μth antenna of multi-antenna transmitter 4.Applying the hopping code inserts a null subcarrier which changesposition, i.e. “hops”, from block to block. Each subcarriercorresponding to a zero symbol is referred to as a null subcarrier.Unlike conventional systems in which training symbols are insertedwithin a single transmission block, each of training symbol insertionunits 15 may insert training symbols over two or more blocks.Consequently, transmitter 4 may insert a sequence of training symbolsover two or more blocks {c_(μ)(k)}_(μ=1) ^(N) ^(t) 14. Sparselyinserting training symbols increases the bandwidth efficiency ofcommunication system 2 because fewer training symbols may be insertedper block {c_(μ)(k)}_(μ=1) ^(N) ^(t) 14. In some embodiments, each oftraining symbol insertion units 15 may insert a particular number oftraining symbols per block {c_(μ)(k)}_(μ=1) ^(N) ^(t) 14 based onchannel 8's coherence time and the pertinent burst duration, e.g. if theburst is long fewer training symbols may be inserted per block{c_(μ)(k)}_(μ=1) ^(N) ^(t) 14. Furthermore, training symbols may beinserted in accordance with existing OFDM standards such as IEEE 802.11aand IEEE 802.11g. Training symbol insertion units 15 are described ingreater detail below using notation introduced in the followingparagraphs.

Subsequent to the insertion of training symbols, MIMO OFDM isimplemented. In particular, each of inverse fast Fourier transform(IFFT) units 17 implement N-point IFFT via left multiplication withF_(N) ^(H) on each corresponding block ū_(μ)(k) 16 and each of cyclicprefix insertion units 19 insert a cyclic prefix via left multiplicationwith the appropriate matrix operator T_(cp):=[I_(L×N) ^(T)I_(N)^(T)]^(T), where I_(L×N) ^(T) represents the last L columns of I_(N).Each of parallel to serial converters (P/S) 21 then parses the resultingblocks {u_(μ)(k)=T_(cp)F_(N) ^(H)ū_(μ)(k)}_(μ=1) ^(N) ^(t) of size P×1into a serial symbol stream. Each of modulators 23 modulate thecorresponding P×1 blocks which are transmitted by the N_(t) transmitantennas over frequency-selective communication channel 8.

Generally, communication channel 8 can be viewed as an L^(th) orderfrequency-selective channel from the μth transmit antenna of transmitter4 to the vth receive antenna of receiver 6. Consequently, communicationchannel 8 can be represented in the discrete-time equivalent formh^((v,μ))(l), lε[0, L] and incorporates transmit and receive filters,g_(μ)(t) and g_(v)(t) respectively, as well as frequency selectivemultipath g_(v,μ)(t), i.e.h^((v,μ))(l)=(g_(μ)*g_(v,μ)*g_(v))(t)|_(t=lT), where * denotesconvolution and T is the sampling period which may be chosen to beequivalent to the symbol period.

Transmissions over communication channel 8 experience a frequencyoffset, f_(o) in Hertz, which may be caused by a mismatch between avoltage controlled oscillator (VCO) of transmitter 4 and a VCO ofreceiver 6 or may also be caused by the Doppler effect. In the presenceof a frequency offset, the samples at with receive antenna can berepresented according to equation (1) below, where ω_(o):=2Πf_(o)T isthe normalized CFO, N_(r) is the number of receive antennas, andη_(v)(n) is zero-mean, white, complex Gaussian distributed noise withvariance σ².

$\begin{matrix}{{{X_{v}(n)} = {{\sum\limits_{\mu = 1}^{N_{t}}{{\mathbb{e}}^{j\; w_{v}n}{\sum\limits_{l = 0}^{L}{{h^{({v,\mu})}(l)}{u_{\mu}\left( {n - 1} \right)}}}}} + {\eta_{v}(n)}}},{v \in \left\lbrack {1,N_{r}} \right\rbrack}} & (1)\end{matrix}$

Each of serial to parallel converters (S/P) 25 convert a respectivereceived sequence x(n) into a corresponding P×1 block 26 with entries[x_(v)(k)]_(p):=x_(v)(kP+p). By selecting block size P greater thanchannel order L each received block x_(v)(k) 26 depends only on twoconsecutive transmitted blocks, u_(μ)(k) and u_(μ)(k−1) which isreferred to as inter-block interference (IBI). In order to substantiallyeliminate IBI at receiver 6, each of cyclic prefix removers 27 removesthe cyclic prefix of the corresponding blocks x_(v)(k) 26 by leftmultiplication with the matrix R_(cp):=[0_(N×L)I_(N)]. The resultingIBI-free block can be represented as y_(v)(k):=R_(cp)x_(v)(k) 28.Equation (2) below can be used to represent the vector-matrixinput-output relationship, where η_(ν)(k):=[η_(v)(kP), η_(v)(kP+1), . .. , η_(v)(kP+P−1)]^(T), with P=N+L; H^((v,μ)) is aP×P lower triangularToeplitz matrix with first column [h^((v,μ))(0), . . . , h^((v,μ))(L),0, . . . , 0]^(T); and D_(P)(ω_(o)) is a diagonal matrix defined asD_(P)(ω_(o)):=diag[1, e^(jω) ^(o) , . . . , e^(jω) ^(o) ^((P-1))].

$\begin{matrix}{{{y_{v}(k)} = {{\sum\limits_{\mu = 1}^{N_{t}}{{\mathbb{e}}^{j\; w_{v}{kP}}R_{cp}{D_{P}\left( w_{o} \right)}H^{({v,\mu})}T_{cp}F_{N}^{H}{{\overset{\_}{u}}_{\mu}(k)}}} + {R_{cp}{\eta_{v}(k)}}}},{v \in \left\lbrack {1,N_{r}} \right\rbrack}} & (2)\end{matrix}$Based on the structure of the matrices involved, it can be readilyverified that R_(cp)D_(p)(w_(ν))=e^(jω) ^(o) ^(L) D_(N)(w_(ν))R_(cp),where D_(N)(w_(ν)):=diag[1, e^(jω) ^(o) , . . . , e^(jω) ^(o) ^((P-1))].Following this identity, we define the N×N matrix {tilde over(H)}^((ν,μ)):=R_(cp)H^((v,μ))T_(cp) as a circulant matrix with firstcolumn [h^((v, μ))(0), . . . , h^((v,μ))(L), 0, . . . , 0]^(T). Lettingalso ν_(v)(k):=R_(cp)η_(v)(k), equation (2) can be rewritten accordingto equation (3).

$\begin{matrix}{{{y_{v}(k)} = {{{\mathbb{e}}^{j\;{w_{v}{({{kP} + L})}}}{D_{N}\left( w_{o} \right)}{\sum\limits_{\mu = 1}^{N_{t}}{{\overset{\sim}{H}}^{({v,\mu})}F_{N}^{H}{{\overset{\_}{u}}_{\mu}(k)}}}} + {v_{v}(k)}}},{v \in \left\lbrack {1,N_{r}} \right\rbrack}} & (3)\end{matrix}$

In the absence of a CFO, taking the FFT of y_(v)(k) 28 renders thefrequency-selective channel 8 equivalent to a set of flat-fadingchannels, since F_(N) ^(H) {tilde over (H)}^((ν,μ)) F_(N) ^(H) is adiagonal matrix D_(N)({tilde over (h)}^((ν,μ))), where {tilde over(h)}^((ν,μ)):=[{tilde over (h)}^((ν,μ))(0), . . . , {tilde over(h)}^((ν,μ))(2Π(N−1/N)]^(T), with

${{\overset{\sim}{h}}^{({v,\mu})}\left( {2\Pi\;{n/N}} \right)}:={\sum\limits_{l = 0}^{L}{{{\overset{\sim}{h}}^{({v,\mu})}(l)}{\exp\left( {{- j}\; 2\pi\;{{nl}/N}} \right)}}}$representing the (ν,μ)th channel's frequency response vales on the FFTgrid. However, in the presence of a CFO, the orthogonality ofsubcarriers is destroyed and the channel cannot be diagonalized bytaking the FFT of y_(v)(k) 28. In order to simplify the input-outputrelationship, F_(N) ^(H)F_(N)=I_(N) can be inserted between D_(N)(w_(o))and {tilde over (H)}^((ν,μ)) to re-express equation (3) as equation (4).

$\begin{matrix}{{{y_{v}(k)} = {{{\mathbb{e}}^{j\;{w_{v}{({{kP} + L})}}}{D_{N}\left( w_{o} \right)}{\sum\limits_{\mu = 1}^{N_{t}}{F_{N}^{H}{D_{N}\left( {\overset{\sim}{h}}^{({v,\mu})} \right)}{{\overset{\_}{u}}_{\mu}(k)}}}} + {v_{v}(k)}}},{v \in \left\lbrack {1,N_{r}} \right\rbrack}} & (4)\end{matrix}$From equation (4) it can be deduced that estimating the CFO and themultiple channels based on {y_(v)(k)}_(v=1) ^(N) ^(r) 28 is a nonlinearproblem. Given {y_(v)(k)}_(v=1) ^(N) ^(r) 28, the CFO ω_(o) and theN_(t)N_(r) channels h^((ν,μ)):=[h^((ν,μ))(0), . . . , h^((ν,μ)) (L)]^(T)in MIMO OFDM communication system 2 are estimated based on the trainingsymbols inserted by training symbol insertion unit 15.

Although ū_(μ)(k) 16 contains both information-bearing symbols andtraining symbols, separation of the information-bearing symbols andtraining symbols is challenging due to the presence of CFO ω_(o). Eachof training symbol insertion units 15 inserts two or more trainingsymbols within the corresponding information-bearing symbolsc_(μ)(k)_(μ=1) ^(N) ^(t) 14 so that CFO estimation can be separated fromMIMO channel estimation. The insertion of the training symbols isperformed in two steps.

In the first step, each of training symbol insertion units 15 inserts ablock of training symbols b_(μ)(k) into the corresponding block ofinformation bearing symbols c_(μ)(k)_(μ=1) ^(N) ^(t) 14 in accordancewith equation (5), where the two permutation matrices P_(A), P_(B) havesizes K×N_(c) and K×N_(b) respectively, and are selected to be mutuallyorthogonal, i.e. P_(A) ^(T), P_(B)=0_(N) _(c) _(×N) _(b) .ũ _(μ)(k)=P _(A) c _(μ)(k)+P _(B) b _(μ)(k)  (5)It is important to note that N_(c)+N_(b)=K and K<N. In some embodiments,P_(A) may be formed with the last N_(c) columns of I_(N) _(c) _(+N) _(b), and P_(B) with the first N_(b) columns of I_(N) _(c) _(+N) _(b) inaccordance with equations (6) and (7), respectively.P _(A) =[e _(N) _(b) . . . e _(K−1)]  (6)P _(A) =[e ₀ . . . e _(N) _(b) ⁻¹]  (7)

The block of training symbols b_(μ)(k) may comprise two or more trainingsymbols and has length N_(b). Moreover, b_(μ)(k) may be one block oftraining symbols in a training sequence including two or more blocks oftraining symbols. By sparsely inserting the training symbols, bandwidthefficiency of communication system 2 can be increased. The resultingstructure of ũ_(μ)(k) in equation (5) is illustrated in FIG. 3. Thestructure of b_(μ)(k) is described in greater detail below.

In the second step, N-K zeros are inserted per block ũ_(μ)(k) to obtainū_(μ)(k). This insertion can be implemented by left-multiplying ũ_(μ)(k)with the hopping code T_(sc) given in equation (8), whereq_(k):=k└N/(L+1)┐.T _(sc)(k):=└e _(qk)(mod N), . . . , e _(qk+K−2(mod N))┘  (8)Applying the hopping code given in equation (8) inserts a zero symbolreferred to as a null subcarrier in each block ũ_(μ)(k). Dependence ofT_(sc) on the block index k implies that the position of the insertednull subcarrier changes from block to block. In other words, equation(8) implements a null subcarrier “hopping” operation from block toblock. By substituting equations (8) and (5) into equation (4) it can bededuced that the resulting signal at the vth receive antenna takes theform of equation (9) given below.

$\begin{matrix}{{y_{v}(k)} = {{\sum\limits_{\mu = 1}^{N_{t}}{{\mathbb{e}}^{j\;{w_{v}{({{k\; P} + L})}}}{D_{N}\left( w_{v} \right)}F_{N}^{H}{D_{N}\left( {\overset{\sim}{h}}_{N}^{v,\mu} \right)}{T_{sc}(k)}{{\overset{\sim}{u}}_{\mu}(k)}}} + {v_{v}(k)}}} & (9)\end{matrix}$

Therefore, each of training symbol insertion units 15 inserts zero andnon-zero training symbols which are used by each of CFO estimators 29and channel estimation unit 33 to estimate the CFO ω_(o) andcommunication channel 8. The null subcarrier is inserted so that theposition of the null subcarrier hops from block to block and enables CFOestimation to be separated from MIMO channel estimation. Consequently,the identifiability of the CFO estimator can be established and theminimum mean square error (MMSE) of the MIMO channel estimator can beachieved.

If CFO ω_(o) was absent, i.e. ω_(o)=0, then the block of trainingsymbols b_(μ)(k) could be separated from the received OFDM transmissionsignal and by collecting the training blocks of a training sequence,communication channel 8 could be estimated using conventionaltechniques. However, the CFO destroys the orthogonality amongsubcarriers of the OFDM transmission signal and the training symbols aremixed with the unknown information-bearing symbols and channels. Thismotivates acquiring the CFO first, and subsequently estimating thechannel.

Each of CFO estimators 29 applies a de-hopping code in accordance withequation (10) on a per block basis.

$\begin{matrix}{{D_{N}^{H}(k)} = {{diag}\left\lbrack {1,{\mathbb{e}}^{{- j}\frac{2\pi}{N}{qk}},\ldots\mspace{14mu},{\mathbb{e}}^{{- j}\frac{2\pi}{N}{{qk}{({N - 1})}}}} \right\rbrack}} & (9)\end{matrix}$Because hopping code T_(sc) is a permutation matrix and D_(N)({tildeover (h)}^((ν,μ))) is a diagonal matrix, it can be verified thatD_(N)({tilde over (h)}^((ν,μ))) T_(sc)(k)=T_(sc)(k) D_(K)({tilde over(h)}^((ν,μ)))(k)), where {tilde over (h)}^((ν,μ)) is formed by permutingthe entries of {tilde over (h)}^((ν,μ)) as dictated by T_(sc)(k). Usingthe de-hopping code given in equation (10), the identity given inequation (11) can be established, where T_(zp):=[I_(K)0_(K×(N−K))] is azero-padding operator.D _(N) ^(H)(k)F _(N) ^(H) T _(sc)(k)=F _(N) ^(H) T _(zp)  (11)

By multiplying equation (9) by the de-hopping code and using equation(11), equation (12) is obtained, where

${g_{v}(k)}:={{\sum\limits_{\mu = 1}^{N_{t}}{{D_{K}\left( {{\overset{\sim}{h}}^{({v,\mu})}(k)} \right)}{{\overset{\sim}{u}}_{\mu}(k)}\mspace{14mu}{and}\mspace{14mu}{{\overset{\_}{v}}_{v}(k)}}}:={{D_{N}^{H}(k)}{{v_{v}(k)}.}}}$y _(ν)(k)=D _(N) ^(H)(k)y _(ν)(k)=e ^(jw) ^(ν) ^((kP+L)) D _(N)(w _(ν))F_(N) ^(H) T _(zp) g(k)+ v _(ν)(k)  (12)

Equation (12) shows that after de-hopping, null subcarriers in differentblocks are at the same location because T_(zp) does not depend on theblock index k.

As a result, the covariance matrix of y _(ν)(k) 30 is given according toequation (13) where the noise v _(ν)(k) has covariance matrix σ²I_(N).R _(yν) =D _(N)(w _(o))F _(N) ^(H) T _(zp) E[g(k)g ^(H)(k)]·T _(zp) ^(H)F _(N) D _(N) ^(H)(w _(ν))+σ² I _(N)  (13)Assuming that the channels are time invariant over M blocks, and theensemble correlation matrix R _(yν) is replaced by its sample estimategiven in equation (14) which is formed by averaging across M blocks,where M>K.

$\begin{matrix}{{\hat{R}}_{y\; v} = {\frac{1}{M}{\sum\limits_{k = 0}^{M - 1}{{{\overset{\_}{y}}_{v}(k)}{{\overset{\_}{y}}_{v}^{H}(k)}}}}} & (14)\end{matrix}$

The column space of R _(yν) has two parts: the signal subspace and thenull subspace. In the absence of CFO, if E[g(k)g^(H)(k)] has full rank,the null space of R _(yν) is spanned by the missing columns, i.e. thelocation of the null subcarriers, of the FFT matrix. However, thepresence of CFO introduces a shift in the null space. Consequently, acost function can be built to measure this CFO-induced phase shift.Representing the candidate CFO as ω, this cost function can be writtenaccording to equation (15), where

$\begin{matrix}{{\sum\limits_{v = 1}^{N_{r}}R_{\overset{\_}{y}v}} = {{{D_{N}\left( w_{o} \right)}F_{N}^{H}T_{zp}\left\{ {\sum\limits_{v = 1}^{N_{r}}{E\left\lbrack {{g(k)}{g^{H}(k)}} \right\rbrack}} \right\} T_{zp}F_{N}{{D_{N}\left( w_{o} \right)}.{J_{V}(\omega)}}}:={\sum\limits_{k = K}^{N - 1}{{f_{N}^{H}\left( \frac{2\pi\; k}{N} \right)}{D_{N}^{- 1}(\omega)}R_{\overset{\_}{y}\; v}{D_{N}(\omega)}{f_{N}\left( \frac{2\pi\; k}{N} \right)}}}}} & (15)\end{matrix}$Consequently, if ω=ω_(o), then D_(N) (ω_(o)−ω)=I_(N). Next, recall thatthe matrix F_(N) ^(H)T_(zp) is orthogonal to {f_(N) (2Πn/N)}_(n=K)^(N−1). Therefore, if ω=ω_(o), the cost function J(ω_(o)) is zero in theabsence of noise. However, for this to be true, ω_(o) must be the uniqueminimum of J(ω). ω_(o) is the unique zero of J(ω) if

$\sum\limits_{v = 1}^{N_{r}}{E\left\lbrack {{g(k)}{g^{H}(k)}} \right\rbrack}$has full rank as established in Proposition 1 below.

$\begin{matrix}{{{{If}\mspace{14mu}{E\left( {{b_{\mu}(k)}{b_{\mu\; H}^{H}(k)}} \right\rbrack}\mspace{14mu}{is}\mspace{14mu}{diagonal}},{\sum\limits_{v = 1}^{N_{r}}{{E\left\lbrack {{b_{\mu}(k)}{b_{\mu\; H}^{H}(k)}} \right\rbrack}\mspace{14mu}{has}\mspace{14mu}{full}\mspace{14mu}{rank}}},{{E\left\lbrack {{c_{\mu}(k)}{c_{\mu\; H}^{H}(k)}} \right\rbrack} = 0},{and}}{{{E\left\lbrack {{b_{\mu\; 1}(k)}{b_{\mu\; 2}^{H}(k)}} \right\rbrack} = 0},{\forall{\mu\; 1}},{\neq {\mu 2}},{then}}{\sum\limits_{v = 1}^{N_{r}}{{E\left\lbrack {{g_{v}(k)}{g_{v}^{H}(k)}} \right\rbrack}\mspace{14mu}{has}\mspace{14mu}{full}\mspace{14mu}{{rank}.}}}} & {{Proposition}\mspace{14mu} 1}\end{matrix}$

Training block b_(μ)(k) satisfies the conditions of proposition 1. Usingthe result of Proposition 1,

$\sum\limits_{v = 1}^{N_{r}}{E\left\lbrack {{g_{v}(k)}{g_{v}^{H}(k)}} \right\rbrack}$has full rank, it follows that J(ω)≧J(ω_(o)), where the equality holdsif and only if ω=ω_(o). Therefore, CFO estimates {circumflex over(ω)}_(o) can be found by minimizing J(ω) according to equation (16).w _(o)=arg_(ω) ^(min) J _(ω)(ω)  (16)

Because of subcarrier hopping, J(ω) has a unique minimum in [−Π, Π)regardless of the position of channel nulls. This establishesidentifiability of {circumflex over (ω)}_(o) and shows that theacquisition range of the CFO estimator given in equation (16) is [−Π,Π), which is the full range.

Based on the CFO estimates produced by equation (16), the terms thatdepend on ω_(o) can be removed from { y _(v)(k)}_(k=0) ^(M−1) 30 andchannel estimation can be performed. In order to derive the MIMO channelestimator, the CFO estimate is temporarily assumed to be perfect, i.e.{circumflex over (ω)}_(o)=ω_(o). After each of CFO estimators 29 removethe CFO related terms from y _(ν)(k) 30, each of FFT units 31 take theFFT of the corresponding block y _(ν)(k) 30 and removes the nullsubcarriers by multiplying the corresponding blocks y _(ν)(k) 30 withT_(zp) ^(T), to obtain z_(ν)(k) 32 according to equation (17), whereξ_(ν)(k):=^(−jω) ^(o) ^((kP+L)) T_(zp) ^(T)F_(N)D_(N) ⁻¹({circumflexover (ω)}_(o)) v _(ν)(k).

$\begin{matrix}\begin{matrix}{{z_{v}(k)} = {{\mathbb{e}}^{{- j}\;{\omega_{o}{({{kP} + L})}}}T_{zp}^{T}F_{N}{D_{N}^{- 1}\left( {\hat{\omega}}_{o} \right)}{{\overset{\_}{y}}_{v}(k)}}} \\{= {{\sum\limits_{\mu = 1}^{N_{t}}{{D_{K}\left( {{\overset{\sim}{h}}^{({v,\mu})}(k)} \right)}\left( {{P_{A}{c_{\mu}(k)}} + {P_{B}{b_{\mu}(k)}}} \right)}} + {\xi_{v}(k)}}}\end{matrix} & (17)\end{matrix}$From the design of P_(A) and P_(B) in equations (6) and (7)respectively, it can be inferred that P_(A) ^(T)D_(K)({tilde over(h)}^((ν,μ))(k))P_(B)=0. This allows the training symbols to beseparated from the received information-bearing symbols in accordancewith equations (18) and (19), where equation (18) represents thereceived information-bearing symbols and equation (19) represents thereceived training symbols.

$\begin{matrix}{{z_{v,c}(k)}:={{P_{A}^{T}{z_{v}(k)}} = {{\sum\limits_{\mu = 1}^{N_{t}}{P_{A}^{T}{D_{K}\left( {{\overset{\sim}{h}}^{({v,\mu})}(k)} \right)}P_{A}{c_{\mu}(k)}}} + {\xi_{v,c}(k)}}}} & (18) \\{{z_{v,b}(k)}:={{P_{B}^{T}{z_{v}(k)}} = {{\sum\limits_{\mu = 1}^{N_{t}}{P_{B}^{T}{D_{K}\left( {{\overset{\sim}{h}}^{({v,\mu})}(k)} \right)}P_{B}{b_{\mu}(k)}}} + {\xi_{v,b}(k)}}}} & (19)\end{matrix}$ξ_(ν,c)(k):=P_(A) ^(T) ξ_(ν)(k) and +ξ_(ν,b)(k):=P_(B) ^(T)ξ_(ν)(k). Bythe definitions of P_(B) in equation (6) and the de-hopping code inequation (11), the identity in equation (20) can be formed, where {tildeover (h)}_(b) ^((ν,μ)) comprises the first N_(b) entries of {tilde over(h)}^((ν,μ)), the N_(b)×(L+1) matrix F (k) comprises the first L+1columns and q_(k) related N_(b) rows of F_(N), andh^((ν,μ)):=[h^((ν,μ))(0), . . . , h^((ν,μ))(L)]^(T).D _(K)({tilde over (h)} ^((ν,μ))(k))P _(B) =P _(B) D _(N) _(b) ({tildeover (h)} _(b) ^((ν,μ))(k))=P _(B)diag[F(k)h ^((ν,μ))]  (20)Because P_(B) ^(T)P_(B)=I_(N) _(b) , equation (20) can be re-expressedaccording to equation (21) where B_(μ)(k):=diag [b_(μ)(k)].

$\begin{matrix}{{z_{v,b}(k)} = {{\sum\limits_{\mu = 1}^{N_{t}}{{B_{\mu}(k)}{F(k)}h^{({v,\mu})}}} + {\xi_{v,b}(k)}}} & (21)\end{matrix}$Note that the length for each block of training symbols, N_(b), can besmaller than N_(t)(L+1) by sparsely distributing training symbols acrossblocks. In some embodiments, N_(t)+1 training symbols are inserted everyN+L transmitted symbols resulting in a bandwidth efficiency of(N−N_(t)−1)/(N+L). Collecting M blocks z_(ν,b)(k), the input-outputrelationship based on training symbols and channels can be expressedaccording to equation (22), where h_(v) comprises {h^((ν,μ))}_(μ=1) ^(N)^(t) , ξ _(ν,b):=[ ξ _(ν,b) ^(T)(0), . . . , ξ _(ν,b) ^(T)(M−1)]^(T),and B is given in equation (23). Note that B is the same for all N_(r)receive antennas

$\begin{matrix}{{\overset{\_}{z}}_{v,b} = {{B\; h_{v}} + {\overset{\_}{\xi}}_{v,b}}} & (22) \\{B = \begin{pmatrix}{{B_{1}(0)}P_{B}^{T}{F(0)}} & \ldots & {{B_{N_{T}}(0)}P_{B}^{T}{F(0)}} \\\vdots & \ddots & \vdots \\{{B_{1}\left( {M - 1} \right)}P_{B}^{T}{F\left( {M - 1} \right)}} & \ldots & {{B_{N_{t}}\left( {M - 1} \right)}P_{B}^{T}{F\left( {M - 1} \right)}}\end{pmatrix}} & (23)\end{matrix}$

By collecting z_(ν,b)'s from all N_(t) transmit antennas into z _(b):=[z _(1,b) ^(T), . . . , z _(N) _(r) _(,b) ^(T)]^(T), the linear MMSE(LMMSE) channel estimator can be expressed according to equation (24),where R_(h):=E[hh^(H)] with h:=[h₁ ^(T), . . . , h_(N) _(r) ^(T)]^(T) asthe channel covariance matrix, and σ² represents the noise variance.ĥ _(LMMSE):=:=(σ² R _(h) ⁻¹ +I _(N) _(r)

(B ^(H) B))⁻¹(I _(N) _(r)

B^(H)) z _(b)  (24)

R_(h) is typically unknown, thus, M N_(b)≧N_(t) (L+1), and B^(H)B isselected to have full rank. In some embodiments, channel estimation unit33 is a least squares (LS) estimator given according to equation (25).ĥ _(LS)=:=(I _(N) _(r)

(B ^(H) B))⁻¹(I _(N) _(r)

B ^(H)) z _(b)  (25)If the number of training symbols per block is N_(b)=N_(t), a minimumnumber of M=L+1 blocks are required to be collected by receiver 6 inorder to guarantee that LS estimation can be performed since h^((ν,μ))with L+1 entries are estimated at the vth receive antenna. In someembodiments, channel estimation unit 33 can be adjusted to collect avariable number of blocks based on the complexity that can be afforded.

The number of b_(μ)(k)'s satisfying the conditions of Proposition 1 isnot unique. For example, N_(b)=N_(t) may be selected and the trainingsequences for different transmit antennas may be designed according toequation (26).b _(μ)(k)=[0_(μ-1) _(r) ^(T) b0_(N) _(t) _(-μ) _(r) ^(T)]^(T)  (26)Further, assume N and M are integer multiples of L+1. Because thehopping step size in equation (8) is N/(L+1), B^(H)B can be designedaccording to equation (27).

                                          (27) $\begin{matrix}{{B^{H}B} = \begin{bmatrix}{\overset{M - 1}{\sum\limits_{m = 0}}{{F^{H}(m)}{B_{1}^{H}(m)}{B_{1}(m)}{F(m)}}} & \; & \; \\\; & \ddots & \; \\\; & \; & {\overset{M - 1}{\sum\limits_{m = 0}}{{F^{H}(m)}{B_{N_{t}}^{H}(m)}{B_{N_{t}}(m)}{F(m)}}}\end{bmatrix}} \\{= {\frac{{b}^{2}M}{N}I_{N_{t}{({L + 1})}}}}\end{matrix}$

Therefore, the number of blocks N improves channel estimationperformance. However, this is true when CFO estimation is perfect. WhenCFO estimation is imperfect, the contrary is true: fewer blocks shouldbe used because the residual CFO estimation error degrades BERperformance when the block index is large.

Thus far, the CFO and N_(t)N_(r) channels have been estimated, but aresidual CFO referred to as phase noise remains. Phase noise degradesthe BER severely as the number of blocks used for channel estimationincreases.

Using the CFO offset {circumflex over (ω)}_(o), produced by each of CFOestimators 29, the received transmission block can be expressedaccording to equation (28) where ({circumflex over (ω)}−ω_(o) is thephase noise and ξ_(ν)(k):=e^(−jω) ^(o) ^((kP+L))T_(zp) ^(T)F_(N)D_(N)⁻¹({circumflex over (ω)}_(o)) v _(ν)(k).{tilde over (y)} _(ν)(k)=e ^(−j(ω) ^(o) ^(−{circumflex over (ω)}) ^(o)^()(kP+L)) D _(N)(ω_(o)−{circumflex over (ω)}_(o))F _(N) ^(H) T _(zp) g_(ν)(k)+ξ_(ν)(k)  (28)When {circumflex over (ω)}_(o) is sufficiently accurate, the matrixD_(N)(ω_(o)−ω_(o)) can be approximated by an identity matrix of the samesize. However, the phase term ({circumflex over (ω)}_(o)−ω_(o))(kP+L)becomes increasingly large as the block index k increases. Withoutmitigating the phase noise, it degrades not only the performance ofchannel estimation unit 33, but also the BER performance over time.

In order to enhance the BER performance, phase estimation unit 35 usesthe non-zero training symbols in b_(μ)(k), which were previouslydesigned to estimate channel 8, to estimate the phase noise per block.For example, assume that for the kth block, the estimated channel isobtained by using the LMMSE channel estimator given in equation (24).Further, also assume that the training sequence is designed as given inequation (26) and that channel estimation is perfect, i.e.D_(N)(ω_(o)−{circumflex over (ω)}_(o))≈I_(N). As a result, afterequalizing channel 8, for the with receive antenna and the μth entry ofz_(ν,b)(k) 30, the equivalent input-output relationship is givenaccording to equation (29), where φ_(v)(k):=[z_(ν,b)(k)]_(μ)/[{tildeover (h)}_(b) ^((ν,μ))]_(μ), and w_(ν) is the equivalent noise termafter removing the channel.φ_(v)(k)=e ^(−j(ω) ^(o) ^(−{circumflex over (ω)}) ^(o) ^()(kP+L)) b+w_(ν)  (29)Because b, is known the phase ({circumflex over (ω)}_(o)−ω_(o))(kP+L)can be estimated based on the observations from N_(r) receive antennason a per block basis. In order to perform this phase estimation step,additional training symbols do not need to be inserted and the extracomplexity is negligible. The performance improvement resulting fromphase estimation is illustrated the performance graphs given below.

After CFO estimation, the FFT has been performed, and channel estimationspace-time decoder 37 decodes the space-time encoded information-bearingsymbols to produce the information-bearing symbol estimates ŝ 38.

Although estimation for a single common CFO and MIMO channel has beendescribed in a single-user system involving N_(t) transmit antennas andN_(r) receive antennas, communication system 2 is not limited to suchsystems. Communication system 2, can easily be modified to estimate CFOsand channel in a multi-user downlink scenario where the base stationdeploys N_(t) transmit antennas to broadcast OFDM based transmissions toN_(r) mobile stations each of which is equipped with one or moreantennas. In this case, there are N_(r) distinct CFOs and N_(t)N_(r)frequency-selective channels to estimate. However, each mobile stationcan still apply perform CFO estimation as given in equation (16). Inaddition, it can be verified that the LS channel estimator given inequation (25) can be separated from CFO estimation to estimate the N_(t)channel impulse responses in h_(v), for v=1, . . . , N_(r), on a perreceive antenna basis.

FIG. 3 illustrates example transmission blocks 40A, 40B, and 40Cgenerated by transmitter 4 of communication system 2. In particular,transmission blocks 40A, 40B, and 40C correspond to consecutivetransmission blocks ū_(μ)(k) 16 at the output of one of training symbolinsertion units 15 with block index k=0, k=1, and k=2, respectively.Generally, each transmission block 40A-40C includes space-time encodedinformation bearing symbols 42A-C, null subcarriers 44A-44C, andtraining symbols 46A-46C, respectively. In particular, training symbolinsertion units 15 insert blocks of N_(t)+1 training symbols 46A-46Caccording to equation (26) as a preamble to space-time encoded blocks ofinformation-bearing symbols 42A-C. In some embodiments, training symbols46A-46C are inserted every N+L transmitted symbols, where a cyclicprefix of L symbols is inserted by cyclic prefix insertion unit 19,resulting in a bandwidth efficiency of (N−N_(t)−1)/(N+L). Additionally,the number of training symbols inserted may be adjusted depending on thechannel's coherence time and the pertinent burst duration

Null subcarriers 44A-44C are inserted within transmission blocks 40A-C,respectively, by applying the hopping code given in equation (8) so thatthe position of null subcarriers 44A-44C change from block to block. Insome embodiments, N−K null subcarriers are inserted with hop-stepN/(L+1) in each transmission block 40A-C. Additionally, null subcarriersmay be inserted in accordance with conventional OFDM standards such asIEEE 802.11a and IEEE 802.11g resulting in easily implemented,low-complexity systems.

FIG. 4 is a flowchart illustrating an example mode of operation ofcommunication system 2 in which receiver 6 performs CFO, channel, andphase noise estimation on an OFDM transmission signal output bytransmitter 4. Generally, transmitter 4 inserts N_(t) training symbolsacross M space-time encoded blocks of information-bearing symbols (step50). The training symbols are inserted as a sequence of blocks oftraining symbols b_(μ)(k) as described herein. In some embodiments, thenumber of training symbols inserted may be adjusted depending on thechannel's coherence time and the pertinent burst duration. Additionally,transmitter 4 may insert two or more training symbols per block ofspace-time encoded information-bearing symbols by applying a first and asecond permutation matrix, P_(A) and P_(B) respectively, as describedpreviously. After inserting the training symbols, transmitter 4 appliesa hopping code to insert N−K null subcarriers per block such that theposition of the null subcarriers changes from block to block (step 52).The hopping code may be defined as in equation (8) with hop-stepN/(L+1). It may be particularly advantageous to insert null subcarriersin accordance with conventional OFDM standards such as IEEE 802.11a andIEEE 802.11g. Transmitter 4 then outputs an OFDM transmission signal byfirst inserting a cyclic prefix and taking the IFFT of the resultingblock of training and information-bearing symbols (step 54).

Receiver 6 receives the OFDM transmission signal and removes the cyclicprefix (step 56). Receiver 6 then applies a de-hopping code andestimates the CFO (step 58). The de-hopping code rearranges the nullsubcarriers so that the null subcarriers in different blocks are at thesame position in their respective blocks, and the CFO is estimated asdescribed previously. Because of the null subcarrier hopping, the CFOestimation and channel estimation can be separated and the CFO can beestimated over the full acquisition range [−Π, Π). The FFT is taken andthe null subcarriers are removed (step 60) by multiplying y _(ν)(k) 30with zero padding matrix T_(zp) ^(T) to obtain z_(v)(k) 32. Channelestimation is performed over M blocks of training symbols (step 62). Asdescribed previously, each training block length N_(b) can be smallerthan N_(t)(L+1) by sparsely distributing training symbols across Mblocks. In some embodiments, one of a LMMSE channel estimator or a LSchannel estimator may be applied to the M blocks to estimate channel 8.In order to improve the BER performance of receiver 6, the phase noiseis estimated and removed (step 64) based on the observations from N_(r)receive antennas on a per block basis. Symbol estimates are thenproduced by decoding the space-time encoded information-bearing symbols(step 66).

FIGS. 5-12 are graphs that present simulations of OFDM transmissionsover MIMO frequency-selective channels using the described techniquesfor estimating the CFO, channel, and phase noise. In order to benchmarkthe performance of the techniques described herein, the Crame'r-Raolower bounds (CRLB) for the CFO are derived. Starting from the model ofcommunication system 2 given in equation (12), the CRLB for ω_(o) isgiven according to equation (30), where D(k):=diag[Pk+L, . . . ,P(k+1)−1], and R_(gg) ^((v)):=E[g_(v)(k)g_(v) ^(H)(k)].

$\begin{matrix}\begin{matrix}{{CRLB}_{\omega} = \left( {\frac{2}{\sigma_{\overset{\_}{v}}^{2}}{\sum\limits_{v = 1}^{N_{r}}{\sum\limits_{k = 0}^{M - 1}{{tr}\left\lbrack {{D(k)}F_{N}^{H}T_{zp}R_{gg}^{(v)}F_{N}{D(k)}} \right\rbrack}}}} \right)^{- 1}} & \;\end{matrix} & (30)\end{matrix}$It follows from equation (30) that as the number of blocks increases,the CRLB for CFO decrease. Similarly, the signal-to-noise ratio (SNR)versus CRLB decreases as the number of blocks increases. If N>>N−K, i.e.the number of subcarriers is much greater than the number of nullsubcarriers, T_(zp)≈I_(N). Assuming that R_(gg) ^((ν))=εI_(N), where εrepresents the average symbol energy, and P, M are sufficiently largeequation (31) can be obtained.

$\begin{matrix}{{CRLB}_{\omega} = {\frac{\sigma_{\overset{\_}{v}}^{2}}{ɛ}\frac{3}{2\left( {P - L} \right)P^{2}M^{3}N_{r}}}} & (31)\end{matrix}$Equation (31) explicitly shows that the CRLB of the CFO is independentof the channel and the number of transmit antennas, and that the CRLB ofthe CFO is inversely proportional to the SNR, the number of receiveantennas, and the cube of the number of space-time data.

By assuming that CFO estimation is perfect, the performance of thechannel estimator can be derived. If the LMMSE channel estimator givenin equation (24) is used, then the mean-square error of the channelestimator is given according to equation (32).

$\begin{matrix}{\sigma_{lmmse}^{2} = {{tr}\left\lbrack \left( {R_{h}^{- 1} + {\frac{M{b}^{2}}{N\;\sigma^{2}}I_{N_{r}{N_{t}{({L + 1})}}}}} \right)^{- 1} \right\rbrack}} & (32)\end{matrix}$Similarly, if the LS channel estimator given in equation (25) is used,the corresponding mean-square error is given by equation (33).

$\begin{matrix}{\sigma_{ls}^{2} = \frac{{NN}_{t}{N_{r}\left( {L + 1} \right)}\sigma^{2}}{M{b}^{2}}} & (33)\end{matrix}$Equations (32) and (33) both imply that as the number of channelsincreases, the channel mean square error increases. However, thisincrease can be mitigated by collecting a greater number of blocks, i.e.more training symbols, provided that the CFO estimate is sufficientlyaccurate.

In all simulations, HIPERLAN/2 channel model B, given in Table 1, isused to generate the channels. The channel order is L=15 and the tapsare independent with different variances. The OFDM block length isdesigned as N=64 as in HIPERLAN/2. The noise is additive white Gaussiannoise with zero-mean and variance σ_(n) ². The SNR is definedSNR=ε/σ_(n) ², and the information-bearing symbols are selected from aquadrature phase-shift keying (QPSK) constellation.

TABLE 1 tap no. 0 1 2 3 4 5 6 7 variance 2.60e−01 2.44e−01 2.24e−017.07e−02 7.93e−02 4.78e−02 2.95e−02 1.78e−02 tap no. 8 9 10 11 12 13 14115 variance 1.07e−02 6.45e−03 5.01e−03 2.51e−03 0 1.48e−03 0 6.02e−04

FIG. 5 is a graph comparing the true frequency offset versus theestimated CFO for the CFO estimation techniques described herein (plot70) and an algorithm described in P. H. Moose, “A technique fororthogonal frequency division multiplexing frequency offset correction,”IEEE Transactions on Communications, vol. 42, pp. 2908-1314, October1994 (plot 72). The ideal line (74) is also shown for comparison andillustrates that the currently described CFO estimation techniques (plot70) has the full acquisition range [−Π,Π), whereas the algorithmdescribed in the P. H. Mooses reference (plot 72) has an acquisitionrange proportional to the OFDM block size N.

FIG. 6 is a graph comparing the effect of the number of blocks overwhich channel estimation is performed for the presently described CFOestimation techniques with N_(t)=2 and N_(r)=2. The CFO is randomlyselected to in the range [−0.5Π, 0.5Π]. In each OFDM transmission block,there are four non-zero training symbols, 4 zero symbols to removeinterference from other channels, and one zero symbol serving as a nullsubcarrier. The placement of the training symbols is in accordance withthe techniques herein, and different numbers of blocks are use: M=L+1(plot 80), M=K (plot 82), 2K (plot 84), 3K (plot 86), and the CRLBderived previously with M=K (plot 88) for comparison. FIG. 6 depicts theCFO normalized mean square error (NMSE), defined as E└∥{circumflex over(ω)}_(o)−ω_(o)∥²/∥ω_(o)∥²┘, verses SNR. As the number of OFDM blocks Mincreases, the NMSE of CFO decreases. However, the improvement isrelatively small, which suggests that using M=K OFDM blocks issufficient to estimate the CFO.

FIG. 7 is a graph comparing the effect of the number of antennas on CFOestimation using the LS channel estimator given in equation (25). Theaverage NMSE of the CFO with the number of blocks M=N are plotted aslines 90, 92, 94, and 96 for systems having (N_(t), N_(r))=(1, 1),(N_(t), N_(r))=(1, 2), (N_(t), N_(r))=(2, 1), (N_(t), N_(r))=(2, 2),respectively. For plots 90 and 92 4 non-zero pilot symbols and one nullsubcarrier per OFDM transmission block are used. FIG. 7 illustrates thatas the number of receive antennas increases, the performance of the CFOestimation techniques described herein increases due to thereceive-diversity gains.

FIG. 8 is a graph comparing the CFO estimation techniques describedherein with a technique described in M. Morelli and U. Mengali, “Animproved frequency offset estimator for OFDM applications,” IEEECommunications Letters, vol. 3, pp. 75-77, March 1999, for the singleantenna case. For the presently described techniques, one non-zerotraining symbol and one zero training symbol for each block are used foreach OFDM transmission block and 64 blocks are collected to perform CFOestimation. In order to maintain the same transmission rate, M.Morelli's and U. Mengali's previously referenced technique has atraining block length of 128 with 8 identical parts. FIG. 8 depicts twocases: random CFO in [−0.06Π, 0.06Π] and fixed CFO with ω_(o)=Π/128. Inboth cases the CFO is chosen within the acquisition range of M. Morelliand U. Mengali's previously referenced technique. In both cases, the CFOtechniques described herein, 100 and 102 for the fixed CFO case and thevarying CFO case, respectively, are comparable with M. Morelli and U.Mengali's technique for the fixed CFO case 104 and varying CFO case 106.

FIG. 9 is a graph comparing the performance of MIMO channel estimationwith (N_(t), N_(r))=(2, 2) and the CFO being randomly selected in therange [−0.5Π, 0.5Π]. By collecting 64 observations from 8 OFDMtransmission blocks and using the LS channel estimator given in equation(25), the MIMO channels can be estimated. In order to measure thechannel estimation quality, the average channel NMSE is computed as E[∥ĥ−h∥²/∥h∥²], where ĥ is obtained using the LS method. The performancefor MIMO OFDM transmissions with estimated CFO 110 using the techniquesdescribed herein are compared with the ideal case in which the CFO isperfectly known 112. FIG. 9 illustrates a 4.5 dB loss due to the CFOestimation error.

FIG. 10 compares the BER performance of the CFO and channel estimationtechniques described herein without phase noise estimation (plot 120),with phase noise estimation (plot 122), and with perfect phase noiseestimation (plot 124) with increasing SNR. The simulation parameters arethe same as those used in FIG. 9 and zero-forcing equalization is usedto estimate the information-bearing symbols. The BER performance of allthe simulations degrades as the number of blocks increase due to phasenoise. As expected, the plot with phase noise estimation 122 performsbetter than the plot without phase noise estimation 120 and the plotwith perfect phase noise estimation 124 provides a benchmark.

FIGS. 11 and 12 compare the estimation of N_(r) CFOs in multi-userbroadcast OFDM systems. Simulations are performed with (N_(t),N_(r))=(2, 2) and CFOs are randomly selected in the range [−0.5Π, 0.5Π].In particular, FIG. 11 illustrates the average channel NMSE with varyingSNRs using a N_(r)×1 vector CFO estimator for the presently describedtechniques with M=L+1 (plot 130), M=K (plot 132), 2K (plot 134), 3K(plot 136). Similarly, FIG. 12 illustrates the BER performance withvarying SNRs using the presently described CFO and channel estimationtechniques without phase noise estimation (plot 140), with phase noiseestimation (plot 142), and with perfect phase noise estimation (plot144). FIGS. 11 and 12 illustrate results which corroborate with FIGS. 9and 10 respectively. Consequently, the described techniques which wereillustrated in detail for a single-user system involving N_(t) transmitantennas and N_(r) receive antennas, can be applied with similar resultsin a multi-user downlink scenario where the base station deploys N_(t)transmit antennas to broadcast OFDM based transmissions to N_(r) mobilestations each of which is equipped with one or more antennas.

Various embodiments of the invention have been described. The inventionprovides techniques for carrier frequency offset (CFO) and channelestimation of orthogonal frequency division multiplexing (OFDM)transmissions over multiple-input multiple-output (MIMO)frequency-selective fading channels. In particular, techniques aredescribed that utilize training symbols in a manner that CFO and channelestimation are decoupled from symbol detection at the receiver. Unlikeconventional systems in which training symbols are inserted within ablock of space-time encoded information-bearing symbols to form atransmission block, the techniques described herein insert trainingsymbols over two or more transmission blocks.

The described techniques can be embodied in a variety of transmittersand receivers used in downlink operation including cell phones, laptopcomputers, handheld computing devices, personal digital assistants(PDA's), and other devices. The devices may include a digital signalprocessor (DSP), field programmable gate array (FPGA), applicationspecific integrated circuit (ASIC) or similar hardware, firmware and/orsoftware for implementing the techniques. If implemented in software, acomputer readable medium may store computer readable instructions, i.e.,program code, that can be executed by a processor or DSP to carry outone of more of the techniques described above. For example, the computerreadable medium may comprise random access memory (RAM), read-onlymemory (ROM), non-volatile random access memory (NVRAM), electricallyerasable programmable read-only memory (EEPROM), flash memory, or thelike. The computer readable medium may comprise computer-readableinstructions that when executed in a wireless communication device,cause the wireless communication device to carry out one or more of thetechniques described herein. These and other embodiments are within thescope of the following claims.

The invention claimed is:
 1. A method comprising: forming two or moreblocks of output symbols for orthogonal frequency division multiplexing(OFDM) transmissions over two or more antennas, wherein the formingcomprises encoding two or more blocks of information-bearing symbols fortransmissions over the two or more antennas, and inserting trainingsymbols and null subcarriers within the two or more blocks ofinformation-bearing symbols at positions determined by a hopping code;and transmitting, via the two or more antennas, transmission signals inaccordance with the two or more blocks of output symbols, wherein thetwo or more blocks of output symbols include a first block of outputsymbols and a second block of output symbols, and wherein transmittingthe transmission signals comprises: transmitting, via a first antenna ofthe two or more antennas, a first transmission signal in accordance withthe first block of output symbols; and transmitting, via a secondantenna of the two or more antennas, a second transmission signal inaccordance with the second block of output symbols.
 2. The method ofclaim 1, wherein transmitting the transmission signals comprisesinserting a cyclic prefix within each of the blocks of output symbols,and wherein the transmission signals provide information for estimatinga carrier frequency offset associated with received versions of thetransmission signals.
 3. The method of claim 2, wherein the transmissionsignals provide information for estimating a phase noise of the receivedversions of the transmission signals based on the estimated carrierfrequency offset.
 4. The method of claim 1, wherein the training symbolswithin the two or more blocks collectively provide information forestimating a channel.
 5. The method of claim 1, wherein inserting thetraining symbols and the null subcarriers comprises inserting at leastone training symbol adjacent to at least one null subcarrier.
 6. Themethod of claim 1, wherein transmitting the transmission signalscomprises transmitting transmission signals for a multi-user wirelesscommunications system.
 7. The method of claim 1, wherein encoding thetwo or more blocks of information-bearing symbols comprises encoding thetwo or more blocks of information-bearing symbols in space and time. 8.A method comprising: encoding information-bearing symbols; forming twoor more blocks of output symbols for orthogonal frequency divisionmultiplexing (OFDM) transmissions over a multiple-input multiple-output(MIMO) channel, wherein the forming comprises inserting training symbolsand null subcarriers within two or more blocks of the encodedinformation-bearing symbols at positions determined by a hopping code;and transmitting, via two or more antennas, transmission signals inaccordance with the two or more blocks of output symbols, wherein themethod further comprises: inserting a cyclic prefix within each of theblocks of output symbols, and applying an inverse fast Fourier transformto the blocks of encoded information-bearing symbols.
 9. A methodcomprising: encoding information-bearing symbols; forming two or moreblocks of output symbols for orthogonal frequency division multiplexing(OFDM) transmissions over a multiple-input multiple-output (MIMO)channel, wherein the forming comprises inserting training symbols andnull subcarriers within two or more blocks of the encodedinformation-bearing symbols at positions determined by a hopping code,wherein the hopping code is a function of an index of at least one blockof the encoded information-bearing symbols; and transmitting, via two ormore antennas, transmission signals in accordance with the two or moreblocks of output symbols.
 10. A method comprising: encodinginformation-bearing symbols; forming two or more blocks of outputsymbols for orthogonal frequency division multiplexing (OFDM)transmissions over a multiple-input multiple-output (MIMO) channel,wherein the forming comprises inserting training symbols and nullsubcarriers within two or more blocks of the encoded information-bearingsymbols at positions determined by a hopping code, wherein the hoppingcode is a function of a property of the MIMO channel; and transmitting,via two or more antennas, transmission signals in accordance with thetwo or more blocks of output symbols.
 11. The method of claim 10,wherein the two or more blocks of output symbols include a first blockof output symbols and a second block of output symbols, and whereininserting the training symbols and null subcarriers comprises: insertinga first null subcarrier at a first subcarrier position within the firstblock of output symbols, inserting a second null subcarrier at a secondsubcarrier position within the second block of output symbols, insertinga first training symbol at a first training symbol position within thefirst block of output symbols, inserting a second training symbol at asecond training symbol position within the second block of outputsymbols, wherein the first subcarrier position is different from thesecond subcarrier position, and wherein the first training symbolposition is different from the second training symbol position.
 12. Themethod of claim 11, wherein the two or more blocks of output symbols areconsecutive, wherein the transmitting the transmission signals comprisestransmitting the transmission signals over a downlink portion of theMIMO channel, and wherein forming the two or more blocks of outputsymbols comprises inserting at least two training symbols and at leasttwo null subcarriers within each block of encoded information-bearingsymbols.
 13. A method comprising: encoding information-bearing symbols;forming two or more blocks of output symbols for orthogonal frequencydivision multiplexing (OFDM) transmissions over a multiple-inputmultiple-output (MIMO) channel, wherein the forming comprises insertingtraining symbols and null subcarriers within two or more blocks of theencoded information-bearing symbols at positions determined by a hoppingcode, wherein the hopping code is a function of the number of antennasfor transmitting the transmission signals; and transmitting, via two ormore antennas, transmission signals in accordance with the two or moreblocks of output symbols.
 14. The method of claim 13, wherein the two ormore blocks of output symbols include a first block of output symbolsand a second block of output symbols, and wherein inserting the trainingsymbols and null subcarriers comprises: inserting a first nullsubcarrier at a first subcarrier position within the first block ofoutput symbols, and inserting a second null subcarrier at a secondsubcarrier position within the second block of output symbols, whereinthe first subcarrier position is different from the second subcarrierposition.
 15. A method comprising: encoding information-bearing symbols;forming two or more blocks of output symbols for orthogonal frequencydivision multiplexing (OFDM) transmissions over a multiple-inputmultiple-output (MIMO) channel, wherein the forming comprises insertingtraining symbols and null subcarriers within two or more blocks of theencoded information-bearing symbols at positions determined by a hoppingcode, wherein inserting the training symbols and the null subcarrierscomprises inserting the training symbols before inserting the nullsubcarriers; and transmitting, via two or more antennas, transmissionsignals in accordance with the two or more blocks of output symbols. 16.A method comprising: encoding information-bearing symbols; forming twoor more blocks of output symbols for orthogonal frequency divisionmultiplexing (OFDM) transmissions over a multiple-input multiple-output(MIMO) channel, wherein the forming comprises inserting training symbolsand null subcarriers within two or more blocks of the encodedinformation-bearing symbols at positions determined by a hopping code;and transmitting, via two or more antennas, transmission signals inaccordance with the two or more blocks of output symbols, wherein thetwo or more blocks of output symbols include a first block of outputsymbols and a second block of output symbols, and wherein transmittingthe transmission signals comprises: transmitting, via a first antenna ofthe two or more antennas, a first transmission signal in accordance withthe first block of output symbols; and transmitting, via a secondantenna of the two or more antennas, a second transmission signal inaccordance with the second block of output symbols.
 17. The method ofclaim 16, wherein transmitting the transmission signals comprisesinserting a cyclic prefix within each of the blocks of output symbols,and wherein the transmission signals provide information for estimatinga carrier frequency offset associated with received versions of thetransmission signals.
 18. The method of claim 16, wherein the trainingsymbols within the two or more blocks collectively provide informationfor estimating the MIMO channel.
 19. The method of claim 16, whereininserting the training symbols and the null subcarriers comprisesinserting at least one training symbol adjacent to at least one nullsubcarrier.
 20. The method of claim 16, wherein transmitting thetransmission signals comprises transmitting transmission signals for amulti-user wireless communications system.
 21. The method of claim 16,wherein inserting the null subcarriers comprises multiplying a block ofencoded information-bearing symbols by a null subcarrier insertionmatrix.
 22. A system comprising: first circuitry configured to (i)encode information-bearing symbols for transmissions over two or moreantennas associated with a multiple-input multiple-output (MIMO) channelto produce encoded information-bearing symbols, and (ii) form two ormore blocks of output symbols for transmissions over the MIMO channel byinserting training symbols and null subcarriers within two or moreblocks of the encoded information-bearing symbols at positionsdetermined by a hopping code; and second circuitry configured totransmit, via the two or more antennas, orthogonal frequency divisionmultiplexing (OFDM) based transmission signals in accordance with thetwo or more blocks of output symbols, wherein the second circuitrycomprises one or more modulators, wherein the two or more blocks ofoutput symbols include a first block of output symbols and a secondblock of output symbols, and wherein the first circuitry is configuredto: insert a first null subcarrier at a first subcarrier position withinthe first block of output symbols, and insert a second null subcarrierat a second subcarrier position within the second block of outputsymbols, wherein the first subcarrier position is different from thesecond subcarrier position.
 23. The system of claim 22, furthercomprising: a wireless communication device configured to receive thetransmission signals as received signals, wherein the wirelesscommunication device is configured to (i) estimate a carrier frequencyoffset based on the received signals and (ii) perform channel estimationof the MIMO channel based on the received signals.
 24. The system ofclaim 22, wherein the first circuitry is configured to: insert a thirdnull subcarrier at a third subcarrier position within the first block ofoutput symbols, insert a fourth null subcarrier at a fourth subcarrierposition within the second block of output symbols, insert a firsttraining symbol at a first training symbol position within the firstblock of output symbols, insert a second training symbol at a secondtraining symbol position within the second block of output symbols,insert a third training symbol at a third training symbol positionwithin the first block of output symbols, and insert a fourth trainingsymbol at a fourth training symbol position within the second block ofoutput symbols, wherein the third subcarrier position is different fromthe fourth subcarrier position, wherein the first training symbolposition is different from the second training symbol position, andwherein the third training symbol position is different from the fourthtraining symbol position.
 25. A system comprising: first circuitryconfigured to (i) encode information-bearing symbols for transmissionsover two or more antennas associated with a multiple-inputmultiple-output (MIMO) channel to produce encoded information-bearingsymbols, and (ii) form two or more blocks of output symbols fortransmissions over the MIMO channel by inserting training symbols andnull subcarriers within two or more blocks of the encodedinformation-bearing symbols at positions determined by a hopping code;and second circuitry configured to transmit, via the two or moreantennas, orthogonal frequency division multiplexing (OFDM) basedtransmission signals in accordance with the two or more blocks of outputsymbols, wherein the second circuitry comprises one or more modulators,wherein the two or more blocks of output symbols include a first blockof output symbols and a second block of output symbols, and wherein thesecond circuitry is configured to: transmit, via a first antenna of thetwo or more antennas, a first transmission signal in accordance with thefirst block of output symbols; and transmit, via a second antenna of thetwo or more antennas, a second transmission signal in accordance withthe second block of output symbols.
 26. The system of claim 25, whereinthe first circuitry is configured to insert a cyclic prefix within eachof the blocks of output symbols, and wherein the transmission signalsprovide information for estimating a carrier frequency offset associatedwith received versions of the transmission signals.
 27. A systemcomprising: first circuitry configured to (i) encode information-bearingsymbols for transmissions over two or more antennas associated with amultiple-input multiple-output (MIMO) channel to produce encodedinformation-bearing symbols, and (ii) form two or more blocks of outputsymbols for transmissions over the MIMO channel by inserting trainingsymbols and null subcarriers within two or more blocks of the encodedinformation-bearing symbols at positions determined by a hopping code,wherein the training symbols within the two or more blocks collectivelyprovide information for estimating the MIMO channel; and secondcircuitry configured to transmit, via the two or more antennas,orthogonal frequency division multiplexing (OFDM) based transmissionsignals in accordance with the two or more blocks of output symbols,wherein the second circuitry comprises one or more modulators.
 28. Thesystem of claim 27, wherein the first circuitry is configured to insertat least one training symbol adjacent to at least one null subcarrier.29. The system of claim 27, wherein each block of output symbolscontains a number of training symbols equal to a number of transmitantennas.